1. **Stating the problem:** We have three polygons arranged side by side: a square, a regular hexagon, and a regular n-sided polygon. We want to understand the relationship between their interior angles or the value of $n$ based on the given information. 2. **Recall the formula for interior angles of a regular polygon:** The measure of each interior angle of a regular polygon with $n$ sides is given by: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ 3. **Known polygons:** - Square: $n=4$, interior angle = $\frac{(4-2) \times 180^\circ}{4} = \frac{2 \times 180^\circ}{4} = 90^\circ$ - Hexagon: $n=6$, interior angle = $\frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ$ 4. **Given:** The sum of the angles around the point where the three polygons meet is $360^\circ$. 5. **Set up the equation:** Let the interior angle of the n-sided polygon be $A_n = \frac{(n-2) \times 180^\circ}{n}$. The sum of the interior angles at the meeting point is: $$90^\circ + 120^\circ + A_n = 360^\circ$$ 6. **Solve for $A_n$:** $$A_n = 360^\circ - 90^\circ - 120^\circ = 150^\circ$$ 7. **Find $n$ using the interior angle formula:** $$150^\circ = \frac{(n-2) \times 180^\circ}{n}$$ Multiply both sides by $n$: $$150n = 180(n - 2)$$ 8. **Simplify:** $$150n = 180n - 360$$ Subtract $180n$ from both sides: $$150n - 180n = -360$$ $$\cancel{150}n - \cancel{180}n = -360$$ $$-30n = -360$$ 9. **Divide both sides by $-30$:** $$\frac{-30n}{-30} = \frac{-360}{-30}$$ $$n = 12$$ **Final answer:** The polygon has $\boxed{12}$ sides.